Orcutt 包使用 OC 程序最后一步的残差计算 DW

Question

This began as a question, but after reading the reference provided in the answer below, as well as the source code, the solution became clear. In case anyone else finds themselves in this position:

orcutt 包，版本 2.2 使用特殊程序计算其 CO 模型的 DW 统计数据。 MWE 使用 orcutt 包的例子来表明它的 Durbin-Watson 统计不是基于 OC 估计的残差。

library(orcutt)
data(icecream, package="orcutt")
dw_calc = function(x){sum((x[2:length(x)] - x[1:(length(x)-1)]) ^ 2) / sum(x ^ 2)}    
lm = lm(cons ~ price + income + temp, data=icecream)
e = lm$residuals
dw_calc(e)
# 1.02117   <- Durbin-Watson Statistic

coch = cochrane.orcutt(lm)
e = coch$residuals
dw_calc(e)
# 1.006431  <- Durbin-Watson Statistic

coch
# Durbin-Watson statistic 
# (original):    1.02117 , p-value: 3.024e-04
# (transformed): 1.54884 , p-value: 5.061e-02

orcutt 包报告 1.54884 但新残差的实际 DW 为 1.006431。报告的值 1.54884 来自收敛过程的最后一轮（参见 Hildreth-Lu）。详见下文：

lm = lm(cons ~ price + income + temp, data=icecream)
reg = lm(cons ~ price + income + temp, data=icecream)
convergence = 8
X <- model.matrix(reg) 
Y <- model.response(model.frame(reg)) 
n<-length(Y) 
e<-reg$residuals   
e2<-e[-1] 
e3<-e[-n] 
regP<-lm(e2~e3-1) 
rho<-summary(regP)$coeff[1] 
rho2<-c(rho) 
XB<-X[-1,]-rho*X[-n,] 
YB<-Y[-1]-rho*Y[-n] 
regCO<-lm(YB~XB-1) 
ypCO<-regCO$coeff[1]+as.matrix(X[,-1])%*%regCO$coeff[-1]   
e1<-ypCO-Y 
e2<-e1[-1] 
e3<-e1[-n] 
regP<-lm(e2~e3-1)  
rho<-summary(regP)$coeff[1] 
rho2[2]<-rho 
i<-2 
while (round(rho2[i-1],convergence)!=round(rho2[i],convergence)){ 
  XB<-X[-1,]-rho*X[-n,] 
  YB<-Y[-1]-rho*Y[-n] 
  regCO<-lm(YB~XB-1) 
  ypCO<-regCO$coeff[1]+as.matrix(X[,-1])%*%regCO$coeff[-1]   
  e1<-ypCO-Y 
  e2<-e1[-1] 
  e3<-e1[-n] 
  regP<-lm(e2~e3-1) 
  rho<-summary(regP)$coeff[1]   
  i<-i+1 
  rho2[i]<-rho 
} 

regCO$number.interaction<-i-1 
regCO$rho <- rho2[i-1]

regCO$DW <- c(lmtest::dwtest(reg)$statistic, lmtest::dwtest(reg)$p.value,
              lmtest::dwtest(regCO)$statistic, lmtest::dwtest(regCO)$p.value)

dw_calc(regCO$residuals)

regF<-lm(YB ~ 1)
tF <- anova(regCO,regF)

regCO$Fs <- c(tF$F[2],tF$`Pr(>F)`[2])

# fitted.value
regCO$fitted.values <- model.matrix(reg) %*% (as.matrix(regCO$coeff))


# coeff
names(regCO$coefficients) <- colnames(X)

# st.err
regCO$std.error <- summary(regCO)$coeff[,2]

# t value
regCO$t.value <- summary(regCO)$coeff[,3]

# p value
regCO$p.value <- summary(regCO)$coeff[,4]       


class(regCO) <- "orcutt"


# formula
regCO$call <- reg$call    

# F statistics and p value
df1 <-  dim(model.frame(reg))[2] - 1
df2 <- length(regCO$residuals) - df1 - 1

RSS <- sum((regCO$residuals)^2)  
TSS <- sum((regCO$model[1] - mean(regCO$model[,1]))^2)

regCO$rse <- sqrt(RSS/df2)
regCO$r.squared <- 1 - (RSS/TSS)
regCO$adj.r.squared <-  1 - ((RSS/df2)/(TSS/(df1 + df2)))

regCO$gdl <- c(df1, df2)

# 
regCO$rank <- df1
regCO$df.residual <- df2    
regCO$assign <- regCO$assign[-(df1+1)]    

regCO$residuals <- Y - regCO$fitted.values
regCO
}

Answer 1

我不太了解这个领域，但似乎更有可能有多种 definitions/estimation 方法用于此统计信息。回到 ?orcutt 中引用的书（Verbeek M. (2004) 现代计量经济学指南，John Wiley & Sons Ltd，ISBN:978-88-08 -17054-5)，并搜索 Google books 得到

上面示例中 orcutt 计算出的值与书中给出的值一致。在本书的前面（第 108 页）它说

In [the Cochrane-Orcutt procedure], $\rho$ and $\beta$ are recursively estimated until convergence, i.e. having estimated $\beta$ by EGLS (by $\beta^*$), the residuals are recomputed and $\rho$ is estimated again using the residuals from the EGLS step. With this new estimate of $\rho$, EGLS is applied again and one obtains a new estimate of $\beta$ ...

换句话说，您上面给出的 $\rho$ 估计值似乎只对应于 Orcutt-Cochrane 程序的第一步。

Orcutt 包使用 OC 程序最后一步的残差计算 DW

The Orcutt Package Calculates DW using residuals from the Last Step of OC Procedure

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